Cosmological Perturbation Theory, 26.4.2011 version

6 PERTURBATIONS IN FOURIER SPACE 13Thus the wavelength λ phys of the Fourier mode ⃗ k grows in time as the universe expands.In the separation into scalar+vector+tensor, we follow Liddle&Lyth and include an additionalfactor k ≡ | ⃗ k| in the Fourier components of B and E i , and a factor k 2 in E, so that wehave, e.g.,B(η,⃗x) = ∑ B ⃗k (η)e i⃗k·⃗x . (6.3)k⃗ kThe purpose of this is to make them have the same dimension and magnitude as B S i , ES ij andE V ij . 12 That is,B i = B S i + B V i , and E ij = E S ij + E V ij + E T ij , (6.4)whereand the conditionsbecomeBi S = −B ,i becomes Bi S = −i k ik BEij S = ( (∂ i ∂ j − 1 3 δ ij∇ 2) E becomes Eij S = − k )ik jk 2 + 1 3 δ ij E ,E V ij = − 1 2 (E i,j + E j,i ) becomes E V ij = − i2k (k iE j + k j E i ), (6.5)δ ij B V i,j = 0, δij E i,j = 0, and δ ik E T ij,k = δij E T ij = 0 (6.6)δ ij k j Bi V = ⃗ k · ⃗B V = 0, δ ij k j E i = ⃗ k · ⃗E = 0, and δ ik k k Eij T = δij Eij T = 0. (6.7)To make the separation into scalar+vector+tensor parts as clear as possible, rotate thebackground coordinates so that the z axis becomes parallel to ⃗ k,(ẑ denoting the unit vector in z direction.) Thenand⎡ ⎤ ⎡10Eij S = 3 E ⎤ ⎡⎣ 0 ⎦ + ⎣ 13 E ⎦ = ⎣1−E3 Eand we can write the scalar part of δg µν as⎡−2Aδgµν S = a2 ⎢ 2(−D + 1 3 E) ⎣2(−D + 1 3 E)+iBFor the vector part we have then⃗ k = kẑ = (0,0,k) (6.8)B S i = (0,0, −iB) (6.9)13 E 13 E − 2 3 E ⎤⎦ (6.10)⎤+iB⎥⎦ (6.11)2(−D − 2 3 E)⃗ k · ⃗ B V = 0 ⇒ ⃗ B V = (B 1 ,B 2 ,0) (6.12)⃗ k · ⃗ E = 0 ⇒ ⃗ E = (E1 ,E 2 ,0) (6.13)12 Powers of k cancel in Eqs. (6.5). The metric perturbations, A, B, D, E, B V i , E i, and E T ij will then all havethe same dimension in Fourier space, which facilitates comparison of their magnitudes.